The Chiral index of the fermionic signature operator

Universit¨ at Regensburg Mathematik

The Chiral index of the fermionic signature operator

Felix Finster

Preprint Nr. 06/2014

arXiv:1404.6625v2 [math-ph] 8 Sep 2015

OPERATOR

FELIX FINSTER APRIL 2014

Abstract. We define an index of the fermionic signature operator on even-dimen- sional globally hyperbolic spin manifolds of finite lifetime. The invariance of the index under homotopies is studied. The definition is generalized to causal fermion systems with a chiral grading. We give examples of space-times and Dirac operators thereon for which our index is non-trivial.

Contents

1. Introduction 1

2. Preliminaries 2

3. The Chiral Index 4

4. Generalization to the Setting of Causal Fermion Systems 5

5. The Chiral Index in the Massless Odd Case 7

6. Homotopy Invariance 7

7. Example: Shift Operators in the Setting of Causal Fermion Systems 9

8. Example: A Dirac Operator with ind₀S6= 0 10

9. Example: A Dirac Operator with indS6= 0 14

10. Examples Illustrating the Homotopy Invariance 18

11. Conclusion and Outlook 20

References 21

1. Introduction

In the recent papers [6, 7] the fermionic signature operator was introduced on glob- ally hyperbolic Lorentzian spin manifolds. It is a bounded symmetric operator on the Hilbert space of solutions of the Dirac equation which depends on the global geometry of space-time. This raises the question how the geometry of space-time is related to spectral properties of the fermionic signature operator. The first step in developing the resulting “Lorentzian spectral geometry” is the paper [5] where the simplest sit- uation of Lorentzian surfaces is considered. In the present paper, we proceed in a somewhat different direction and show that there is a nontrivial index associated to the fermionic signature operator. This is the first time that an index is defined for a geometric operator on a Lorentzian manifold.

We make essential use of the decomposition of spinors in even space-time dimension into left- and right-handed components (the “chiral grading”). The basic idea is to

Supported in part by the Deutsche Forschungsgemeinschaft.

1

decompose the fermionic signature operator S using the chiral grading as S=S_L+S_R with S^∗

L=S_R, (1.1)

and to define the so-calledchiral indexofSas the Noether index ofS_L. After providing the necessary preliminaries (Section 2), this definition will be given in Section 3 in space-times of finite lifetime. In order to work out the mathematical essence of our index, in Section 4 we also give its definition in the general setting of causal fermion systems (for an introduction to causal fermion systems see [3] or [4]). Section 5 is devoted to a variant of the chiral index which applies in the special case of the massless Dirac equation and a Dirac operator which is odd with respect to the chiral grading.

In Section 6 we analyze the invariance properties of the chiral indices when space-time or the Dirac operator are deformed by a homotopy. In Sections 8–10 we construct examples of fermionic signature operators with a non-trivial index and illustrate the homotopy invariance. Finally, in Section 11 we discuss our results and and give an outlook on potential extensions and applications, like the generalization to space-times of infinite lifetime.

We point out that the purpose of this paper is to define the chiral index, to study a few basic properties and to show in simple examples that it is in general non-trivial.

But we do not work out any physical applications, nor do we make the connection to geometric or topological invariants. These shortcomings are mainly due to the fact that we only succeeded in computing the index explicitly in highly symmetric and rather artificial examples. Moreover, it does not seem easy to verify the conditions needed for the homotopy invariance. For these reasons, we leave physically interesting examples and geometric stability results as a subject of future research. All we can say for the moment is that the chiral index describes the “chiral asymmetry” of the Dirac operator in terms of an integer. This integer seems to depend on the geometry of the boundary of space-time and on the singular behavior of the potentials in the Dirac equation. Smooth potentials in the Dirac equation, however, tend to not affect the index.

2. Preliminaries

We recall a few basic constructions from [6]. Let (M, g) be a smooth, globally hyperbolic Lorentzian spin manifold of even dimension k ≥ 2. For the signature of the metric we use the convention (+,−, . . . ,−). We denote the spinor bundle by SM. Its fibersS_xM are endowed with an inner product≺.|.≻_x of signature (n, n) withn= 2^k/2−1(for details see [1, 8]), which we refer to as the spin scalar product. Clif- ford multiplication is described by a mapping γ which satisfies the anti-commutation relations,

γ : TxM_→L(SxM_{) with γ(u)γ(v) +γ(v)γ(u) = 2g(u, v) 11S}

x(M₎. We write Clifford multiplication in components with the Dirac matricesγ^j and use the short notation with the Feynman dagger, γ(u) ≡ u^jγ_j ≡ /u. The metric connections on the tangent bundle and the spinor bundle are denoted by ∇.

In the even-dimensional situation under consideration, the spinor bundle has a de- composition into left- and right-handed components. We describe this chiral grading by an operator Γ (the “pseudoscalar operator,” in physics usually denoted by γ⁵),

Γ : S_xM_→SxM_,

having for all u∈T_xMthe properties

Γ^∗=−Γ, Γ² = 11, Γγ(u) =−γ(u) Γ, ∇Γ = 0 (2.1) (where the star denotes the adjoint with respect to the spin scalar product). We denote the chiral projections to the left- and right-handed components by

χL= 1

2 11−Γ

and χR= 1

2 11 + Γ

. (2.2)

The sections of the spinor bundle are also referred to as wave functions. We denote the smooth sections of the spinor bundle by C^∞(M_{, S}M). Similarly, C₀^∞(M_{, S}M₎ denotes the smooth sections with compact support. On the compactly supported wave functions, one can introduce the Lorentz invariant inner product

<.|.> : C₀^∞(M_{, S}M_)×C^∞

0 (M_{, S}M_{)→C, (2.3)}

<ψ|φ>:=

Z

M

≺ψ|φ≻_xdµM. (2.4)

The Dirac operator D is defined by

D:=iγ^j∇_j+B : C^∞(M_{, S}M_)→C^∞₍M_{, S}M_),

where B ∈ L(Sx) (the “external potential”) typically is a smooth multiplication op- erator which is symmetric with respect to the spin scalar product. In some of our examples,Bwill be chosen more generally as a convolution operator which is symmet- ric with respect to the inner product (2.4). For a given real parameter m ∈ R (the

“mass”), the Dirac equation reads

(D −m)ψ= 0.

We mainly consider solutions in the class C_sc^∞(M_{, S}M) of smooth sections with spa- tially compact support. On such solutions one has the scalar product

(ψ|φ) = 2π Z

N

≺ψ|/νφ≻_xdµN(x), (2.5)

whereN denotes any Cauchy surface andν its future-directed normal. Due to current conservation, the scalar product is independent of the choice of N (for details see [6, Section 2]). Forming the completion gives the Hilbert space (H_m,(.|.)).

For the construction of the fermionic signature operator, we need to extend the bilinear form (2.4) to the solution space of the Dirac equation. In order to ensure that the integral in (2.4) exists, we need to make the following assumption (for more details see [6, Section 3.2]).

Definition 2.1. A globally hyperbolic space-time(M_{, g)} is said to bem-finite if there is a constant c >0 such that for all φ, ψ∈H_m∩Csc^∞(M_{, S}M), the function ≺φ|ψ≻_x is integrable on M _and

|<φ|ψ>| ≤ckφk kψk (where k.k= (.|.)¹² is the norm on H_m).

Under this assumption, the space-time inner product is well-defined as a bounded bilinear form on H_m,

<.|.> : H_m×H_m→C.

Applying the Riesz representation theorem, we can uniquely represent this bilinear form with a signature operator S,

S : H_m →H_m with <φ|ψ>= (φ|Sψ). (2.6) We refer to S as the fermionic signature operator. It is obviously a bounded symmetric operator onH_m. We note that the construction of the fermionic signature operator is manifestly covariant and independent of the choice of a Cauchy surface.

3. The Chiral Index

We now modify the construction of the fermionic signature operator by inserting the chiral projection operators into (2.4). We thus obtain the bilinear forms

<ψ|φ>_L/R = Z

M

≺ψ|χ_L/Rφ≻_xdµM. (3.1) For the space-time integrals to exist, we need the following assumption.

Definition 3.1. A globally hyperbolic space-time (M_{, g)} is said to beΓ-finite if there is a constant c >0such that for all φ, ψ∈H_m∩Csc^∞(M_{, S}M), the function ≺φ|Γψ≻_x is integrable on M _and

|<φ|Γψ>| ≤ckφk kψk.

There seems no simple relation between m-finiteness and Γ-finiteness. But both con- ditions are satisfied if we assume that the space-time (M_{, g) has} finite lifetime in the sense that it admits a foliation (N_t)t∈(t

0,t1) by Cauchy surfaces with t₀, t₁ ∈ R such that the function hν, ∂_tiis bounded on M(see [6, Definition 3.4]). The following proposition is an immediate generalization of [6, Proposition 3.5].

Proposition 3.2. Every globally hyperbolic manifold of finite lifetime is m-finite and Γ-finite.

Proof. Let ψ∈H_m∩C_sc^∞(M_{, S}M_{) and} C(x) one of the operators 11_Sx or iΓ(x). Ap- plying Fubini’s theorem and decomposing the volume measure, we obtain

<ψ|Cψ>= Z

M

≺ψ|Cψ≻(x)dµM(x) = Z t1

t0

Z

N_t

≺ψ|Cψ≻ hν, ∂_tidt dµN_t

and thus

<ψ|Cψ>

≤sup

M

hν, ∂_ti Z t1

t0

dt Z

N_t

|≺ψ|Cψ≻|dµN_t. Rewriting the integrand as

|≺ψ|Cψ≻|=|≺ψ|/ν(/νC)ψ≻|,

the bilinear form ≺.|/ν.≻is a scalar product. Moreover, the operator /νC is symmetric with respect to this scalar product. Using that

(/ν)² = 11 = (i/νΓ)²,

we conclude that the sup-norm corresponding to the scalar product ≺.|/ν.≻ of the operator νC/ is equal to one. Hence

Z

N_t

|≺ψ|Cψ≻|dµN_t ≤ Z

N_t

≺ψ|/νψ≻dµN_t = (ψ|ψ),

and consequently

<ψ|Cψ>

≤(t₁−t₀) sup

M

hν, ∂_ti kψk².

Polarization and a denseness argument give the result.

Assuming that our space-time is m-finite and Γ-finite, the bilinear forms (3.1) are bounded onH_m×H_m. Thus we may represent them with respect to the Hilbert space scalar product in terms of signature operators S_L/R,

SL/R : H_m→H_m with <φ|ψ>_L/R = (φ|S

L/Rψ). (3.2) We refer to S_L/R as the chiral signature operators. Taking the complex conjugate of the equation in (3.2) and using that χ^∗_L =χR, we find that (1.1) holds, where the star denotes the adjoint in L(H_m).

We now define the chiral index as the Noether index ofS_L (sometimes called Fred- holm index; for basics see for example [9, §27.1]).

Definition 3.3. The fermionic signature operator is said to have finite chiral index if the operators of S_L and S_R both have a finite-dimensional kernel. The chiral index of the fermionic signature operator is defined by

indS= dim kerS_L−dim kerS_R. (3.3) 4. Generalization to the Setting of Causal Fermion Systems Our starting point is a causal fermion system as introduced in [3].

Definition 4.1. Given a complex Hilbert space (H,h.|.iH) (the“particle space”) and a parametern∈N(the“spin dimension”), we letF⊂L(H) be the set of all self-adjoint operators on H of finite rank, which (counting with multiplicities) have at most n positive and at most nnegative eigenvalues. On F we are given a positive measureρ (defined on a σ-algebra of subsets of F), the so-called universal measure. We refer to (H,F, ρ) as a causal fermion system.

Starting from a Lorentzian spin manifold, one can construct a corresponding causal fermion system by choosingHas a suitable subspace of the solution space of the Dirac equation, forming the local correlation operators (possibly introducing an ultraviolet regularization) and definingρas the push-forward of the volume measure onM_{(see [6,} Section 4] or the examples in [4]). The advantage of working with a causal fermion system is that the underlying space-time does not need to be a Lorentzian manifold, but it can be a more general “quantum space-time” (for more details see [2]).

We now recall a few basic notions from [3]. OnFwe consider the topology induced by the operator norm kAk:= sup{kAukH withkukH= 1}. For every x∈F we define thespin spaceS_x byS_x=x(H); it is a subspace ofHof dimension at most 2n. OnS_x we introduce thespin scalar product ≺.|.≻_x by

≺u|v≻x=−hu|xuiH (for all u, v ∈Sx) ; (4.1) it is an indefinite inner product of signature (p, q) withp, q≤n. Moreover, we define space-time M as the support of the universal measure, M = suppρ. It is a closed subset of F.

In order to extend the chiral grading to causal fermion systems, we assume for every x∈M an operator Γ(x)∈L(H) with the properties

Γ(x)|_Sx : S_x→S_x and xΓ(x) =−Γ(x)^∗x . (4.2)

We define the operators χ_L/R(x) ∈ L(H) again by (2.2). In order to explain the equations (4.2), we first note that the right side of (4.2) obviously vanishes on the orthogonal complement ofS_x. Using furthermore that, by definition of the spin space, the operator x is invertible on S_x, we infer that

Γ(x)|_S⊥ x = 0. Moreover, the computation

≺ψ|Γ(x)φ≻_x=−hψ|xΓ(x)φiH=−hΓ(x)^∗x ψ|φiH (4.2)

= hxΓ(x)ψ|φiH=−≺Γ(x)ψ|φ≻x

(with ψ, φ ∈ S_x) shows that Γ(x) ∈ L(S_x) is antisymmetric with respect to the spin scalar product. Thus the first equation in (2.1) again holds. This implies that the adjoint of χ_L(x) with respect to≺.|.≻_xequals χ_R(x). However, we point out that our assumptions (4.2) do not imply that Γ(x) is idempotent (in the sense that Γ(x)²|_Sx = 11_Sx). Hence the analog of the second equation in (2.1) does not need to hold on a causal fermion system. This property could be imposed in addition, but will not be needed here. The last two relations in (2.1) do not have an obvious correspondence on causal fermion systems, and they will also not be needed in what follows.

We now have all the structures needed for defining the fermionic signature operator and its chiral index. Namely, replacing the scalar product in (2.6) by the scalar product on the particle space h.|.iH, we now demand in analogy to (2.4) and (2.6) that the relation

hu|SviH = Z

M

≺u|v≻_xdρ(x)

should hold for allu, v ∈H. Using (4.1), we find that the fermionic signature operator is given by the integral

S=− Z

M

x dρ(x).

Similarly, the left-handed signature operator can be introduced by S_L=−

Z

M

x χ_Ldρ(x). (4.3)

In the setting on a globally hyperbolic manifold, we had to assume that the manifold was m-finite and Γ-finite (see Definitions 2.1 and 3.1). Now we need to assume cor- respondingly that the integral (4.3) converges. For the sake of larger generality we prefer to work with weak convergence.

Definition 4.2. The topological fermion system isS_L-boundedif the integral in (4.3) converges weakly to a bounded operator, i.e. if there is an operator S_L ∈ L(H) such that for all u, v ∈H,

− Z

M

hu|x χ_LviHdρ(x) =hu|S_LviH. Introducing the right-handed signature operator byS_R:=S^∗

L, we can define thechiral indexagain by (3.3).

5. The Chiral Index in the Massless Odd Case

We return to the setting of Section 3 and consider the special case that the mass vanishes and that the Dirac operator is odd,

m= 0 and ΓD=−DΓ. (5.1)

In this case, the solution space of the Dirac equation is obviously invariant under Γ, Γ : H₀→H₀.

Taking the adjoint with respect to the scalar product (2.5) and noting that Γ anti- commutes withν, one sees that Γ is symmetric on/ H₀. Henceχ_Landχ_Rare orthogonal projection operators, giving rise to the orthogonal sum decomposition

H₀=H_L⊕H_R with H_L/R :=χ_L/RH₀. (5.2) Moreover, the computation

<χ_Lψ|χ_c′φ>_c = Z

M

≺χ_Lψ|χ_cχ_c′φ≻_xdµM

= Z

M

≺ψ|χ_Rχ_cχ_c′φ≻_xdµM=δ_Rcδ_cc′ <ψ|φ>_c

with c, c^′ ∈ {L, R} (and similarly for L replaced by R) shows that S maps the right- handed component to the left-handed component and vice versa. Moreover, in a block matrix notation corresponding to the decomposition (5.2), the operators S_L and S_R have the simple form

S_L=

0 0

A 0

and S_R=

0 A^∗

0 0

with a bounded operator A :H_L→ H_R. As a consequence, bothS_L and S_R have an infinite-dimensional kernel, so that the index cannot be defined by (3.3). This problem can easily be cured by restricting the operators to the respective subspaceH_LandH_R. Definition 5.1. In the massless odd case (5.1), the fermionic signature operator is said to have finite chiral index if the operators S_L|H_L and S_R|H_R both have a finite- dimensional kernel. We define the index ind0Sby

ind0S= dim ker(S_L)|H_L−dim ker(S_R)|H_R. 6. Homotopy Invariance

We first recall Dieudonn´e’s general theorem on the homotopy invariance of the Noether index (see for example [9, Theorem 27.1.5”]).

Theorem 6.1. Let T(t) :U → V, 0 ≤t ≤1, be a one-parameter family of bounded linear operators between Banach spaces U and V which is continuous in the norm topology. If for every t∈[0,1] the vector spaces

kerT andV /T(H) are both finite-dimensional, (6.1) then

indT(0) = indT(1), where indT := dim ker(T)−dimV /T(H).

In most applications of this theorem, one knows from general arguments that the index of T remains finite under homotopies (for example, in the prominent example of the Atiyah-Singer index, this follows from elliptic estimates on a compact manifold).

For our chiral index, however, there is no general reason why the chiral index ofSshould remain finite. Indeed, the fermionic signature operator is bounded and typically has many eigenvalues near zero. It may well happen that for a certain value oft, an infinite number of these eigenvalues becomes zero (for an explicit example see Example 10.1 below).

Another complication when applying Theorem 6.1 to the fermionic signature opera- tor is that the image of S_Ldoes not need to be a closed subspace of our Hilbert space.

To explain the difficulty, we first consider the chiral index of Definition 3.3. Using that kerS_R= kerS^∗

L =S_L(H_m)^⊥, the assumption that the fermionic signature opera- tor has finite chiral index can be restated that the vector spaces kerS_L and S_L(H_m)^⊥ are finite-dimensional subspaces of H_m. Since

dimS_L(H_m)^⊥= dimH_m/ S_L(H_m) ,

this implies that the closure of the image of S_L has finite co-dimension. If the image of S_L were closed in H_m, the finiteness of the chiral index would imply that the conditions (6.1) hold if we set T =S_L and U =V =H_m. However, the image of S_L will in general notbe a closed subspace of H_m, and in this case it is possible that the condition (6.1) is violated for T =S_L and U =V =H_m, although Shas finite chiral index (according to Definition 3.3). In the massless odd case, the analogous problem occurs if we choose T =S_L,U =H_L and V =H_R (see Definition 5.1).

Our method for making Theorem 6.1 applicable is to endow a subspace of the Hilbert space with a finer topology, such that the image ofS_Llies in this subspace and is closed in this topology.

Theorem 6.2. Let S(t) : H_m → H_m, t ∈ [0,1], be a family of fermionic signature operators with finite chiral index. Let E be a Banach space together with an embed- ding ι:E ֒→H_m with the following properties:

(i) For everyt∈[0,1], the image ofS_L(t) lies in ι(E), giving rise to the mapping

S_L(t) : H_m→E . (6.2)

(ii) For every t ∈ [0,1], the image of the operator S_L, (6.2), is a closed subspace of E.

(iii) The family S_L(t) :H_m →E is continuous in the norm topology.

Then the chiral index is a homotopy invariant, indS(0) = indS(1).

In the chiral odd case, the analogous result is stated as follows.

Theorem 6.3. Let S(t) : H₀ → H₀, t ∈ [0,1], be a family of fermionic signature operators of finite chiral index in the massless odd case (see (5.1)). Moreover, letE be a Banach space together with an embedding ι:E ֒→H_R such that the operatorS_L|H_L : H_L→H_R has the following properties:

(i) For everyt∈[0,1], the image ofS_L(t) lies in ι(E), giving rise to the mapping

S_L(t) : H_L→E . (6.3)

(ii) For every t ∈ [0,1], the image of the operator S_L, (6.3), is a closed subspace of E.

(iii) The family S_L(t) :H_L→E is continuous in the norm topology.

Then the chiral index in the massless odd case is a homotopy invariant, ind₀S(0) = ind₀S(1).

In Example 10.2 below, it will be explained how these theorems can be applied.

7. Example: Shift Operators in the Setting of Causal Fermion Systems In the remainder of this paper we illustrate the previous constructions in several examples. The simplest examples for fermionic signature operators with a non-trivial chiral index can be given in the setting of causal fermion systems. We let H=ℓ²(N) be the square-summable sequences with the scalar product

hu|viH= X∞

l=1

u_lv_l.

For anyk∈Nwe define the operators x_k by

(x_ku)_k =−u_k+1, (x_ku)_k+1=−u_k,

and all other components of x_ku vanish. Thus, writing the series in components, x_ku= ( 0, . . . ,0

| {z }

k−1 entries

,−u_k+1,−u_k,0, . . .). (7.1) Every operator x_k obviously has rank two with the non-trivial eigenvalues ±1. We let µ be the counting measure on N and ρ =x_∗(µ) the push-forward measure of the mapping x :k 7→x_k ∈F ⊂L(H). We thus obtain a causal fermion system (H,F, ρ) of spin dimension one.

Next, we introduce the pseudoscalar operators Γ(x_k) by Γ(x_k)u= ( 0, . . . ,0

| {z }

k−1 entries

, u_k,−u_k+1,0, . . .). (7.2) Obviously, these operators have the properties (4.2). Moreover,

x χ_L(x_k)u= (0, . . . ,0, −u_k+1, 0, 0, . . .) x χ_R(x_k)u= (0, . . . ,0, 0, −u_k, 0, . . .). Consequently, the operators

S_L/R =− X∞ k=1

x χ_L(x_k) (7.3)

take the form

S_Lu= (u₂, u₃, u₄, . . .), S_Ru= (0, u₁, u₂, . . .)

(note that the series in (7.3) converges weakly; in fact it even converges strongly in the sense that the series P

k(x_kχ_Lu) converges in H for everyu ∈H). These are the usual shift operators, implying that

indS= 1.

We finally remark that a general index p ∈ Ncan be arranged by modifying (7.1) and (7.2) to

x_ku= ( 0, . . . ,0

| {z }

k−1 entries

,−u_k+p,0, . . . ,0

| {z }

p−1 entries

, −u_k, 0, . . .) Γ(x_k)u= ( z }| {

0, . . . ,0 , u_k, z }| {

0, . . . ,0, −u_k+p,0, . . .).

Moreover, a negative index can be arranged by exchanging the left- and right-handed components.

8. Example: A Dirac Operator with ind0S6= 0

We now construct a two-dimensional space-time (M, g) together with an odd Dirac operator D such that the resulting fermionic signature operator in the massless case has a non-trivial chiral index ind₀ (see Definition 5.1). We choose M_{= (0,2π)×S}¹ with coordinates t∈(0,2π) and ϕ∈[0,2π). We begin with the flat Lorentzian metric

ds²=dt²−dϕ². (8.1)

We consider two-component complex spinors, with the spin scalar product

≺ψ|φ≻=hψ|

0 1 1 0

φi_C2. (8.2)

We choose the pseudoscalar matrix as Γ =

−1 0

0 1

, (8.3)

so that

χ_L= 1 0

0 0

, χ_R=

0 0 0 1

. (8.4)

The space-time inner product (2.4) becomes

<ψ|φ>= Z 2π

0

Z 2π 0

≺ψ(t, ϕ)|φ(t, ϕ)≻dϕ dt . (8.5) The Dirac operator Dshould be chosen to be odd (see the right equation in (5.1)).

This means that D has the matrix representation D=

0 D_R DL 0

(8.6) with suitable operators DL and DR. In order for current conservation to hold, the Dirac operator should be symmetric with respect to the inner product (8.5). This implies that the operators D_L and D_R must both be symmetric,

D^∗_L=D_L, D_R^∗ =D_R, (8.7)

where the star denotes the formal adjoint with respect to the scalar product on the Hilbert spaceL²(M_,C). We consider the massless Dirac equation

Dψ= 0. (8.8)

The scalar product (2.5) on the solutions takes the form (ψ|φ) = 2π

Z _2π

0

hψ(t, ϕ)|φ(t, ϕ)i_C2 dϕ , (8.9)

bc bc

k ω

b

b

b

b

b b

b

bb

b b b

b

ω_−1,L 1 ω_−2,L

1 ω1,R

ω_2,L ω_2,R

ω_−2,R

Figure 1. The eigenvalues ω_k,L/R in the case p= 2.

giving rise to the Hilbert space (H₀,(.|.)). As a consequence of current conservation, this scalar product is independent of the choice of t.

We assume that the system is invariant under time translations and is a first order differential operator in time. More precisely, we assume that

D_L/R=i∂t−H_L/R (8.10)

with purely spatial operators H_L/R, referred to as the left- and right-handed Hamilto- nians. Moreover, we assume that these Hamiltonians are homogeneous. This implies that they can be diagonalized by plane waves,

D_ce^ikϕ=ω_k,ce^ikϕ with k∈Z and c∈ {L, R}.

As a consequence, the Dirac equation (8.8) can be solved by the plane waves e_k,L = 1

2π e^{−iωk,Lt+ikϕ} 1

0

, e_k,R = 1

2π e^{−iωk,Rt+ikϕ} 0

1

. (8.11) The vectors (e_k,c)k∈Z,c∈{L,R} form an orthonormal basis of the Hilbert space H₀. We remark that the Dirac operator of the Minkowski vacuum is obtained by choosing

H_L=i∂_ϕ, H_R=−i∂_ϕ

(see for example [5] or [4, Section 7.2]). In this case, ω_k,L/R = ∓k. More generally, choosing D_c as a homogeneous differential operator of first order, the eigenvalues ω_k,c are linear in k. Here we do not want to assume that the operators D_c are differ- ential operators. Then the eigenvalues ω_k,L and ω_k,R can be chosen arbitrarily and independently, except for the constraint coming from the symmetry (8.7) that these eigenvalues must be real.

More specifically, for a given parameter p∈N we choose ω_k,L =−k and ω_k,R=

k if k≤0

k+p if k >0 (8.12) (see Figure 1). Then the space-time inner product of the basis vectors (e_k,c)k∈Z,c∈{L,R}

is computed by

<e_k,L|e_k′,L>= 0 =<e_k,R|e_k′,R>

<e_k,R|e_k′,L>= 1 2π δ_k,k^′

Z _2π

0

e^{i(ωk,R−ωk,L)t}dt=δ_k,k^′δ_{ωk,R, ωk,L} =δ_k,0δ_k^′_,0.

We conclude Sdoes not have finite chiral index.

In order to obtain a non-trivial index, we need to modify our example. The idea is to change the space-time inner product in such a way that the inner product between two different plane-wave solutions with the same frequencies becomes non-zero. As a consequence, the corresponding pair of plane-wave solutions will disappear from the kernel. The only vectors which remain in the kernel are those which do not have a partner for pairing, so that

kerS_L|H_L = span e_−1,L, . . . ,e_−p,L

, kerS_R|H_R ={0}.

(see again Figure 1, where the pairs are indicated by horizontal dashed lines, whereas the vectors in the kernel correspond to the circled dots). Generally speaking, the method to modify the space-time inner product for states with the same frequency is to insert a potential into the Dirac equation which is time-independent but has a non-trivial spatial dependence. It is most convenient to work with a conformal transformation. Thus we go over from the Minkowski metric (8.1) to the conformally flat metric

d˜s² =f(ϕ)² dt²−dϕ²

, (8.13)

wheref ∈C^∞(R/(2πZ)) is a non-negative, smooth, 2π-periodic function. The confor- mal invariance of the Dirac equation (for details see for example [4, Section 8.1] and the references therein) states in our situation that the Dirac operator transforms as

D˜ =f⁻³² Df¹² , (8.14)

so that

D˜ =

0 D˜_R D˜_L 0

with D˜_L/R =f⁻³²D_L/Rf¹² .

The solutions of the massless Dirac equation are modified simply by a conformal factor,

ψ˜=f⁻¹² ψ . (8.15)

The space-time inner product (8.5) and the scalar product (8.9) transform to

<ψ|˜φ>˜ = Z 2π

0

Z 2π 0

≺ψ(t, ϕ)˜ |φ(t, ϕ)≻˜ f(ϕ)²dϕ dt

= Z _2π

0

Z _2π

0

≺ψ(t, ϕ)|φ(t, ϕ)≻f(ϕ)dϕ dt (8.16) ( ˜ψ|φ) = 2π˜

Z 2π 0

hψ(t, ϕ)|˜ φ(t, ϕ)i˜ _C2 f(ϕ)dϕ

= 2π Z _2π

0

hψ(t, ϕ)|φ(t, ϕ)i_C² dϕ= (ψ|φ). (8.17) To understand these transformation laws, one should keep in mind that the spin scalar product remains unchanged under conformal transformations. The same is true for the integrand ≺ψ|/νφ≻x of the scalar product (2.5), because the operator ν/ is normalized by /ν² = 11.

From (8.17) we conclude that the scalar product does not change under conformal transformations. In particular, the conformally transformed plane-wave solutions

˜e_k,L/R=f(ϕ)⁻¹² e_k,L/R (8.18)

are an orthonormal basis of ˜H₀. The space-time inner product (8.16), however, involves a conformal factor f(ϕ). As a consequence, the space-time inner product of the basis vectors (˜e_k,c)k∈Z,c∈{L,R} can be computed by

<˜e_k,L|˜e_k′,L>= 0 =<˜e_k,R|˜e_k′,R>

<˜e_k,R|˜e_k′,L>= Z 2π

0

dt Z 2π

0

f(ϕ)dϕ≺˜e_k,R(t, ϕ)|˜e_k′,L(t, ϕ)≻

= 1

2π δωk,R, ω_k′,L

Z 2π 0

f(ϕ)e^{−i(k−k′)ϕ}dϕ= 1

2π δωk,R,ω_k′,L

fˆ_k−k^′ , where ˆf_k is thek^th Fourier coefficient off,

f(ϕ) = 1 2π

X

k∈Z

fˆ_ke^ikϕ.

Using the explicit form of the frequencies (8.12), we obtain the following invariant subspaces and corresponding matrix representations of S,

Sˆ|_span(˜e_−k,L,˜e_k,R)= 1 2π

0 fˆ_2k fˆ_2k 0

!

if k≤0 ˆS|_span(˜e_−k−p,L,˜e_k,R)= 1

2π

0 fˆ_2k+p fˆ_2k+p 0

!

if k > p ˆS|_span(˜e_−1,L,˜e_−p,L)= 0.

In particular, we can read off the chiral index:

Proposition 8.1. Assume that almost all Fourier coefficients fˆ_k of the conformal function in (8.13)are non-zero. Then the fermionic signature operator in the massless odd case has finite chiral index (see Definition 5.1) and ind₀S=p.

We finally compute the Dirac operator in position space. The dispersion relations in (8.12) are realized by the operators

D_L=i(∂_t−∂_ϕ) D_R=i(∂_t+∂_ϕ) +B, whereB is the spatial integral operator

Bψ

(t, ϕ) = Z 2π

0

B(ϕ, ϕ^′)ψ(t, ϕ^′)dϕ^′ with the distributional integral kernel

B(ϕ, ϕ^′) =− p 2π

X∞ k=1

e^{ik(ϕ−ϕ′)}=−p

2δ(ϕ−ϕ^′)− p 2π

PP e^{−i(ϕ−ϕ′)}−1. Hence, choosing the Dirac matrices as

γ⁰ = 0 1

1 0

, γ¹ =

0 1

−1 0

(8.19) and using (8.4), we obtain

D=iγ⁰∂_t+iγ¹∂_ϕ+γ¹χ_RB. (8.20)

Performing the conformal transformation (8.14), we finally obtain Dψ)(t, ϕ) =˜ i

f(ϕ)

γ⁰∂_t+γ¹∂_ϕ+ f^′(ϕ) 2f(ϕ) −p

2 γ¹χ_R

ψ(t, ϕ) (8.21)

− p 2π

γ¹χ_R f(ϕ)³²

Z _2π

0

PP

e^{−i(ϕ−ϕ′)}−1 ψ(t, ϕ^′)p

f(ϕ^′)dϕ^′. (8.22) Thus (8.21) is the Dirac operator in the Lorentzian metric (8.13) with a constant right-handed potential. Moreover, the summand (8.22) is a nonlocal integral operator involving a singular integral kernel.

This example shows that the index of Proposition 8.1 in general does not encode the topology of space-time, because for a fixed space-time topology the index can take any integer value. The way we understand the index is that it gives topological information on the singular behavior of the potential in the Dirac operator.

9. Example: A Dirac Operator with indS6= 0

We now construct an example of a fermionic signature operator for which the in- dex indSof Definition 3.3 is non-trivial. To this end, we want to modify the example of the previous section. The major difference to the previous setting is that the Hilbert space H_m does not have a decomposition into two subspaces H_L and H_R, making it necessary to consider the operators S_L and S_R as operators on the whole solution space H_m. Our first task is to remove the infinite-dimensional kernels of the oper- ators S_L and S_R. This can typically be achieved by perturbing the Dirac operator, for example by introducing a rest mass. The second and more substantial modifica- tion is to arrange that the operators S_L and S_R have infinite-dimensional invariant subspaces. This is needed for the following reason: In the example of the previous section, the operatorS_L|H_L:H_L→H_R mapped one Hilbert space to another Hilbert space. Therefore, we obtained a non-trivial index simply by arranging that the op- erator S_L|H_L gives a non-trivial “pairing” of vectors of H_L with vectors of H_R (as indicated in Figure 1 by the horizontal dashed lines). In particular, if considered as an operator onH₀, the operatorS_Lhad at most two-dimensional invariant subspaces.

For the chiral index of Definition 3.3, however, we have only one Hilbert space H_m to our disposal, so that the operator S_L : H_m → H_m is an endomorphism of H_m. As a consequence, the chiral index is trivial whenever H_m splits into a direct sum of finite-dimensional subspaces which are invariant underS_L (because on each invariant subspace, the index is trivial due to the rank-nullity theorem of linear algebra).

The following example is designed with the aim of showing in explicit detail that the index is non-zero. Our starting point are the plane-wave solutions (8.11) with the frequencies according to (8.12) with p ∈ N. In Figure 2 the transformed plane-wave solutions ˜e_k,c (where the transformation frome_k,c to ˜e_k,c will be explained below) are arranged according to their frequencies and momenta on a lattice. We shall construct the operator S_L in such a way that these plane-wave solutions are mapped to each other as indicated by the arrows. Thus similar to a shift operator, S_L maps the basis vectors to each other “spiraling in,” implying that the vector ˜e_−1,L (depicted with the circled dot) is in the kernel of S_L. Likewise, the operatorS_Racts like a “spiraling out”

shift vector, so that it is injective. In this way, we arrange that indS= 1. Similarly, in the case p >1 we shall obtain p spirals, so that indS=p.

bc

k ω

b

b

b

b b

b

bb b

b

1

˜e_−3,L

1

˜e_2,L

˜ e_2,R

Figure 2. The action of S_L on the transformed plane-wave solutions in the case p= 1.

Before entering the detailed construction, we point out that our method is driven by the wish that the example should be explicit and that the kernels of the chiral signature operators should be given in closed form. This makes it necessary to introduce a Dirac operator which seems somewhat artificial. In particular, instead of introducing a rest mass, we arrange a mixing of the left- and right-handed components using a time-dependent vectorial gauge transformation. Moreover, we again work with a conformal transformation with a carefully adjusted spatial and time dependence. We consider these special features merely as a requirement needed in order to make the computations as simple as possible. In view of the stability result of Theorem 6.2, we expect that the index is also non-trivial in more realistic examples involving a rest mass and less fine-tuned potentials. But probably, this goes at the expense of longer computations or less explicit arguments.

We begin on the cylinderM_{= (0,6π)×S}¹, again with the Minkowski metric (8.1) and two-component spinors endowed with the spin scalar product (8.2). The space- time inner product (2.4) becomes

<ψ|φ>= Z 6π

0

Z 2π 0

≺ψ(t, ϕ)|φ(t, ϕ)≻dϕ dt , (9.1) whereas the scalar product on solutions of the Dirac equation is again given by (8.9).

We again consider the massless Dirac equation (8.8) with the Dirac operator (8.6) and the left- and right-handed operators according to (8.10). Moreover, we again assume that the operators D_L/R have the plane-wave solutions (8.11) with frequencies (8.12).

For a fixed real parameter ν 6= 0, we consider the transformation U(t) = 11 +iνγ⁰cost/3

p1 +ν²cos²t/3 = 1

p1 +ν²cos²t/3

1 iνcost/3 iνcost/3 1

. (9.2) Obviously,U(t)∈U(2) is a unitary matrix. Moreover, it commutes withγ⁰, implying that it is also unitary with respect to the spin scalar product. As a consequence, the transformation U(t) is unitary both on the Hilbert space H₀ and with respect to the inner product (9.1). Next, we again consider a conformal transformation (8.14) and (8.15), but now with a conformal function f(t, ϕ) which depends on space and time. Thus we set

D˜ =f⁻³²UDU^∗f¹² and ψ˜=f⁻¹² U ψ . (9.3)

b b

V L =

L , R R ω

ω+¹₃ ω− ¹₃

V ^b = ^b R L R L

Figure 3. The transformationV in momentum space.

Similar to (8.16) and (8.17), the inner products transform to

<ψ|˜φ>˜ = Z 6π

0

Z 2π 0

≺ψ(t, ϕ)|φ(t, ϕ)≻f(t, ϕ)dϕ dt and ( ˜ψ|φ) = (ψ|φ)˜ . In particular, the transformed plane wave solutions ˜e_k,care an orthonormal basis ofH₀. Keeping in mind that the chiral projectors in (3.1) donotcommute withU, we obtain

<ψ|˜φ>˜ _L= Z 6π

0

Z 2π 0

≺U(t)ψ(t, ϕ)|χ_LU(t)φ(t, ϕ)≻f(t, ϕ)dϕ dt and thus, in view of (3.2),

(˜e_k,c|S_L˜e_k′,c^′) = Z _6π

0

Z _2π

0

≺Ue_k,c|χ_LUe_k′,c^′≻f(t, ϕ)dϕ dt . (9.4) In order to get rid of the square roots in (9.2), it is most convenient to set

V(t) =

1 iνcost/3 iνcost/3 1

and µ(t, ϕ) = f(t, ϕ)

1 +ν²cos²t/3. (9.5) Then (9.4) simplifies to

(˜e_k,c|S_L˜e_k′,c′) = Z 6π

0

Z 2π 0

≺Ve_k,c|χ_LVe_k′,c′≻µ(t, ϕ)dϕ dt . (9.6) Let us first discuss the effect of the transformation V. A left-handed spinor is mapped to

V 1

0

= 1

0

+ i 2 e^it/3

0 1

+ i 2 e^−it/3

0 1

.

Thus two right-handed contributions are generated, whose frequency differ from the frequency of the left-handed component by ±1/3. Similarly, a right-handed spinor is mapped to

V 0

1

= 0

1

+ i 2 e^it/3

1 0

+ i 2 e^−it/3

1 0

,

generating two left-handed components with frequencies shifted by ±1/3. Again plot- ting the frequencies vertically, we depict the transformation V as in Figure 3. The same notation is also used in Figure 2 for the transformed plane-wave solutions.

The inner product ≺.|χL.≻ in (9.6) only gives a contribution if the arguments on the left and right have the opposite chirality. Since the transformed plane-wave solu- tionsV e_k,c have a fixed chirality at every lattice point, one sees in particular that (9.6) vanishes if µ is chosen as a constant. By adding to the constant µ= 1 contributions with different momenta, we can connect the different lattice points in Figure 2. This leads us to the ansatz

µ(t, ϕ) = 1 +µ_hor(t, ϕ) +µ_vert(t, ϕ), (9.7)

where the last two summands should describe the horizontal respectively vertical ar- rows in Figure 2. For the horizontal arrows we can work similar to (8.18) with a spatially-dependent conformal transformation. However, in order to make sure that the left-handed component generated by V (corresponding to the two Ls at the very right of Figure (2)) are not connected horizontally, we include two Fourier modes which shift the frequency by ±2/3,

µ_hor(t, ϕ) =a(ϕ)

1−e^2it3 −e^−2it3

, (9.8)

where ahas the Fourier decomposition a(ϕ) =

X∞

k=1

a_ke^ikϕ+a_−ke^−ikϕ

. (9.9)

For the vertical arrows we must be careful that the left-handed contribution ofVe_k,L is not connected to the right-handed component ofVe_k,R, because then the arrow would have the wrong direction. To this end, we avoid integer frequencies. Instead, we work with the frequencies inZ±1/3, because they connect the left-handed componentVe_k,R to the right-handed component of Ve_k,L. This leads us to the ansatz

µ_vert(t, ϕ) =µ_vert(t) =X

n∈Z

e^int

b_ne^it3 +b_−ne^−it3

. (9.10)

The ans¨atze (9.9) and (9.10) ensure that µ is real-valued. Moreover, by choosing the Fourier coefficients sufficiently small, one can clearly arrange that the first summand in (9.7) dominates, so that µis strictly positive. We thus obtain the following result.

Proposition 9.1. Assume that the Fourier coefficients a_k and b_n in (9.9) and (9.10) are sufficiently small and that almost all Fourier coefficients are non-zero. Then the function µ defined by (9.8) and (9.7), is strictly positive. Consider the Dirac oper- ator (9.3) with U and f according to (9.2) (for some fixed ν ∈ R\ {0}) and (9.5).

Then the chiral index of the fermionic signature operator (see Definition 3.3) is finite and indS=p.

We finally discuss the form of the Dirac operator in position space. Substitut- ing (8.20) into (9.3) and using the above form of U and f, the Dirac operator ˜D can be computed in closed form. Similar as discussed in the previous section, the Dirac operator contains a nonlocal integral operator with a singular potential. Moreover, the transformation U modifies the Dirac matrix γ¹ to

γ¹ →U γ¹U^∗ = 1 1 +ν²cos²(t/3)

1−ν²cos²(t/3)

γ¹−2νcos²(t/3)γ² , where

γ²:=

i 0 0 −i

.

Thus the representation of the Dirac matrices becomes time-dependent; this is the main effect of the vectorial transformation U. This transformation changes the first order terms in the Dirac equation. Moreover, the conformal transformation also changes the first order terms just as in (8.21) by a prefactor 1/f.

10. Examples Illustrating the Homotopy Invariance

We now give two examples to illustrate our considerations on the homotopy invari- ance of the chiral index. We begin with an example which shows that the dimension of the kernel ofS_Ldoes not need to be constant for deformations which are continuous in L(H₀). It may even become infinite-dimensional.

Example 10.1. We consider the space-time M _{= (0, T)×S}¹ with coordinates t ∈ (0, T) and ϕ∈[0,2π) endowed with the Minkowski metric

ds²=dt²−dϕ².

We again choose two-component complex spinors with the spin scalar product (8.2).

The Dirac operator is chosen as

D=iγ⁰∂_t+iγ¹∂_ϕ,

where the Dirac matrices are again given by (8.19). The pseudoscalar matrix and the chiral projectors are again chosen according to (8.3) and (8.4).

We consider the massless Dirac equation Dψ= 0.

This equation (8.8) can be solved by plane wave solutions, which we write as e_k,L(ζ) = 1

2π e^+ikt+ikϕ 1

0

, e_k,R(ζ) = 1

2π e^−ikt+ikϕ 0

1

, (10.1)

where k ∈ Z(the indices L and R denote the left- and right-handed components; at the same time they propagate to the left respectively right). By direct computation, one verifies that (e_k,c)k∈Z,c∈{L,R} is an orthonormal basis of H₀.

We next compute the space-time inner product (2.4),

<e_k,R|e_0,L>= Z T

0

dt Z 2π

0

dϕ≺e_k,R(t, ϕ)|e_0,L(t, ϕ)≻= 1 2π

Z T 0

δ_k,0dt= T 2π δ_k,0

<e_k,R|e_k′,L>= 1 2π δ_k,k^′

Z T 0

e^2iktdt= e^2ikT −1

4πik (k^′ 6= 0)

<e_k,L|e_k′,L>= 0 =<e_k,R|e_k′,R> .

Thus the fermionic signature operator S is invariant on the subspaces H^(k)

0 gener- ated by the basis vectors e_k,L and e_k,R. Moreover, in these bases it has the matrix representations

S|_H(0) 0

= T 2π

0 1 1 0

and S|_H(k) 0

= 1

4πik

0 e^2ikT −1 e^−2ikT −1 0

(k6= 0). If T 6∈ πQ, the matrix entries e^±2ikT −1 are all non-zero. As a consequence, the operators S_L|H_L and S_R|H_R are both injective. Thus Shas finite chiral chiral index in the massless odd case (see Definition 5.1). IfT ∈πQ, however, the chiral index vanishes for all k for which 2kT is a multiple of 2π. As a consequence, the operators S_L|H_L

and S_R|H_R both have an infinite-dimensional kernel, so that S does not have a finite

chiral index. ♦

This example also explains why we need additional assumptions like those in The- orems 6.2 and 6.3. In particular, when considering homotopies of space-time or of the

Dirac operator, one must be careful to ensure that the chiral index remains finite along the chosen path.

We next want to construct examples of homotopies to which the stability result of Theorem 6.3 applies. To this end, it is convenient to work similar to (8.13) with a conformal transformation.

Example 10.2. As in Example 10.1 we consider the space-time (0, T)×S¹, but now with the conformally transformed metric

d˜s² =f(t)² dt²−dϕ² wheref is a non-negativeC²-function with

suppf ⊂(−T, T) and f(0)>0.

Similar to (8.18) and the computation thereafter, transforming the plane-wave solu- tions (10.1) conformally to ˜e_k,L/R =f(t)⁻¹²e_k,L/R, we again obtain an orthonormal basis of H₀ and

<˜e_k,R|˜e_k′,L>= 1 2π δ_k,k^′

Z T 0

f(t)e^2iktdt

<˜e_k,L|˜e_k′,L>= 0 =<˜e_k,R|˜e_k′,R>

for all k, k^′ ∈Z.

The integration-by-parts argument Z T

0

f(t)e^2iktdt= 1 2ik

Z T 0

f(t) d

dte^2iktdt=−f(0) 2ik − 1

2ik Z T

0

f^′(t)e^2iktdt

=−f(0)

2ik − f^′(0) 4k² − 1

4k² Z T

0

f^′′(t)e^2iktdt

shows that the space-time inner products have a simple explicit asymptotics for largek given by

<˜e_k,R|˜e_k′,L>= f(0)

4πik δ_k,k^′+O 1 k²

. Hence the operator S_L has the form

S_Le_k,L=c_ke_k,R with coefficients c_k having the asymptotics

c_k= f(0)

4πik +O 1 k²

.

From this asymptotics we can read off the following facts. First, it is obvious thatS_L|H_L

has a finite-dimensional kernel. Exchanging the chirality, the same is true for S_R|H_R, implying thatShas a finite chiral index (according to Definition 5.1). Next, the vectors in the image ofS_L are in the Sobolev spaceW^1,2,

S_L|H_L : H_L→H_R∩W^1,2(S¹,C²).

Moreover, the image of this operator is closed (in theW^1,2-norm). Finally, our partial integration argument also yields that

kS_Lψk_W1,2 ≤ |f|_C2 kψkH₀ ,

showing that the family of signature operators is norm continuous for aC²-homotopy of functionsf.

Having verified the assumptions of Theorem 6.3, we conclude that the chiral index in the massless odd case is invariant underC²-homotopies of the conformal functionf,

provided that f(0) stays away from zero. ♦

11. Conclusion and Outlook

Our analysis shows that the chiral index of a fermionic signature operator is well- defined and in general non-trivial. Moreover, it is a homotopy invariant provided that the additional conditions stated in Theorems 6.2 and 6.3 are satisfied. As already mentioned at the end of the introduction, the physical and geometric meaning of this index is yet to be explored.

We now outline how our definition of the chiral index could be generalized or ex- tended other situations. First, our constructions also apply in the Riemannian setting by working instead of causal fermion systems with so-called Riemannian fermion sys- tems or general topological fermion systemsas introduced in [4]. In this situation, one again imposes a pseudoscalar operators Γ(x) ∈L(H) with the properties (4.2). Then all constructions in Section 4 go through. Starting on an even-dimensional Riemannian spin manifold, one can proceed as explained in [4] and first construct a corresponding topological fermion system. For this construction, one must choose a particle space, typically of eigensolutions of the Dirac equation. Once the topological fermion system is constructed, one can again work with the index of Section 4. If the Dirac operator anti-commutes with the pseudoscalar operator, one can choose the particle space Hto be invariant under the action of Γ. This gives a decomposition of the particle space into two chiral subspaces,H =H_L⊕H_R. Just as explained in Section 5, this makes it possible to introduce other indices by restricting the chiral signature operators to H_L or H_R. Moreover, one could compose the operators from the left with the projection operators onto the subspaces H_L/R and consider the Noether indices of the resulting operators.

Another generalization concerns space-times of infinite lifetime. Using the con- structions in [7], in such space-times one can still introduce the fermionic signature operator S_m provided that the space-time satisfies the so-called mass oscillation prop- erty. By inserting chiral projection operators, one can again define chiral signature operators S^L/R_m and define the chiral index as their Noether index. Also the stability results of Theorems 6.2 and 6.3 again apply. It is unknown whether the resulting in- dices have a geometric meaning. SinceS_m depends essentially on the asymptotic form of the Dirac solutions near infinity, the corresponding chiral indices should encode information on the metric and the external potential in the asymptotic ends.

We finally remark that the fermionic signature operator could be localized by re- stricting the space-time integrals to a measurable subset Ω ⊂ M. For example, one can introduce a chiral signature operator S_L(Ω) similar to (3.1) and (3.2) by

(φ|S_L(Ω)ψ) = Z

Ω

≺ψ|χLφ≻xdµM.

Likewise, in the setting of causal fermion systems, one can modify (4.3) to S_L(Ω) =−

Z

Ω

x χ_Ldρ(x).